Problem 41 - Fundamentals of Mathematics

Question

Due to a slowdown in the industry, a Silicon Valley computer company finds itself in debt \(\$ 2,400,000\). Use absolute value notation to describe this company's debt.

Step-by-Step Solution

Answer

\(|-2,400,000|\)

Step 1: Understand the Context

The company is in debt, indicating a negative financial situation. The debt amount is \( \\(2,400,000 \). In terms of mathematics, this is represented with a negative sign as \( -\\)2,400,000 \).

Step 2: Apply Absolute Value Definition

The absolute value of a number is its distance from zero on the number line, disregarding its sign. Therefore, the absolute value of \(-\$2,400,000\) is simply the magnitude of the debt, which discards the negative sign.

Step 3: Use Absolute Value Notation

The absolute value is represented using vertical bars around a number. In this case, the absolute value notation for the company's debt is \( |-2,400,000| \).

Key Concepts

Absolute Value Notation

When we talk about absolute value notation, we're referring to a way of expressing a number's distance from zero on the number line, regardless of whether that number is positive or negative. It's like measuring how far away something is, without caring about the direction. We use vertical bars to denote this concept. For example, if you see the expression \(|-2,400,000|\), this is asking for the absolute value of -2,400,000.

• The notation \(|x|\) means "the absolute value of x."
• No matter how negative the number inside the bars is, the result is always positive or zero.
• Absolute value is critical in real-world situations where only the size or magnitude of a number matters, not its sign.

Thus, applying absolute value notation to debt or any financial loss simply signifies the magnitude of the deficit, helping to clarify the extent or size of the loss without focusing on the negative aspect itself.

Negative Numbers

Negative numbers are numbers less than zero, usually represented with a minus sign in front of them. They are interesting because they describe quantities or values below zero, which can be counterintuitive in everyday experiences where zero is traditionally the lowest point. We encounter them in various scenarios such as temperatures below freezing, elevations below sea level, and of course, financial debts.

• Negative numbers are used to represent a decrease or a debt in financial contexts.
• On a number line, negative numbers are always to the left of zero.
• Financially, they often show that something is owed, like a debt.

In the given exercise, the company's debt of \\(2,400,000 is shown as -\\)2,400,000. This negative sign is an important detail, which signals that this amount is owed or not currently possessed by the company.

Debt Representation

Debt representation is a way to illustrate financial obligations where a person or company owes money. In mathematical terms, it's often shown as a negative figure because it signifies an amount yet to be settled or paid off. When someone is in debt, it's crucial to understand what this represents in absolute terms as well.

• Debt is essentially money that is borrowed and needs to be returned.
• Mathematically, it is often indicated by a negative number.
• Understanding debt through absolute value helps in grasping the total amount owed without the negative connotations.

In the context of our exercise, the company's debt of \(-2,400,000\) is precisely a representation of how much they need to pay back. Using absolute value notation, \(|-2,400,000|\) simplifies this understanding by focusing on the total liability without dwelling on the negative sign. This approach assists in financial planning and reporting by concentrating solely on the size of the debt.